Shafarevich-Tate groups of abelian varieties

Abstract

The Shafarevich-Tate group $W(\mathcal{A})$ measures the failure of the Hasse principle for an abelian variety $\mathcal{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W(\mathcal{A})\cong Cl(\Lambda )\oplus Cl(\Lambda )$ or $W(\mathcal{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right)\oplus C{l}_{\mathbf{odd}}(\Lambda )\oplus C{l}_{\mathbf{odd}}(\Lambda )$, where $Cl(\Lambda )$ is the ideal class group of a ring $\Lambda$ associated to the K-theory of the non-commutative tori and $2^k$ divides the order of $Cl(\Lambda )$. The case of elliptic curves with complex multiplication is considered in detail.

12 pages

References in corpus (2)

• The Cassels-Tate pairing on polarized abelian varieties
• Comments on a note by M. Waldschmidt

Cited by in corpus (1)

• Quantum dynamics of elliptic curves